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Properties

Label 117.2.ba.b
Level $117$
Weight $2$
Character orbit 117.ba
Analytic conductor $0.934$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,2,Mod(71,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.71");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.ba (of order \(12\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{24}^{7} q^{2} + \zeta_{24}^{2} q^{4} + ( - \zeta_{24}^{5} + 2 \zeta_{24}^{3} - \zeta_{24}) q^{5} + ( - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{4}) q^{7} + (3 \zeta_{24}^{5} - 3 \zeta_{24}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{7} q^{2} + \zeta_{24}^{2} q^{4} + ( - \zeta_{24}^{5} + 2 \zeta_{24}^{3} - \zeta_{24}) q^{5} + ( - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{4}) q^{7} + (3 \zeta_{24}^{5} - 3 \zeta_{24}) q^{8} + (2 \zeta_{24}^{6} - \zeta_{24}^{4} + \cdots + 2) q^{10}+ \cdots + ( - 4 \zeta_{24}^{7} + \cdots - 4 \zeta_{24}^{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{7} + 12 q^{10} - 28 q^{13} - 4 q^{16} - 16 q^{19} - 8 q^{22} + 8 q^{28} - 16 q^{31} + 12 q^{34} + 44 q^{37} - 48 q^{40} + 24 q^{43} + 24 q^{46} - 48 q^{49} + 16 q^{55} + 4 q^{58} + 4 q^{61} + 8 q^{67} + 40 q^{70} + 44 q^{73} + 16 q^{76} + 16 q^{79} - 24 q^{82} - 40 q^{91} - 32 q^{94} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(\zeta_{24}^{2}\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
0.965926 − 0.258819i
−0.965926 + 0.258819i
−0.965926 + 0.258819i 0 −0.866025 + 0.500000i −2.63896 − 2.63896i 0 1.00000 − 3.73205i 2.12132 − 2.12132i 0 3.23205 + 1.86603i
71.2 0.965926 − 0.258819i 0 −0.866025 + 0.500000i 2.63896 + 2.63896i 0 1.00000 − 3.73205i −2.12132 + 2.12132i 0 3.23205 + 1.86603i
80.1 −0.258819 + 0.965926i 0 0.866025 + 0.500000i 0.189469 + 0.189469i 0 1.00000 − 0.267949i −2.12132 + 2.12132i 0 −0.232051 + 0.133975i
80.2 0.258819 − 0.965926i 0 0.866025 + 0.500000i −0.189469 − 0.189469i 0 1.00000 − 0.267949i 2.12132 − 2.12132i 0 −0.232051 + 0.133975i
89.1 −0.965926 − 0.258819i 0 −0.866025 − 0.500000i −2.63896 + 2.63896i 0 1.00000 + 3.73205i 2.12132 + 2.12132i 0 3.23205 − 1.86603i
89.2 0.965926 + 0.258819i 0 −0.866025 − 0.500000i 2.63896 − 2.63896i 0 1.00000 + 3.73205i −2.12132 − 2.12132i 0 3.23205 − 1.86603i
98.1 −0.258819 − 0.965926i 0 0.866025 − 0.500000i 0.189469 − 0.189469i 0 1.00000 + 0.267949i −2.12132 − 2.12132i 0 −0.232051 − 0.133975i
98.2 0.258819 + 0.965926i 0 0.866025 − 0.500000i −0.189469 + 0.189469i 0 1.00000 + 0.267949i 2.12132 + 2.12132i 0 −0.232051 − 0.133975i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.ba.b ✓ 8
3.b odd 2 1 inner 117.2.ba.b ✓ 8
13.c even 3 1 1521.2.i.f 8
13.e even 6 1 1521.2.i.c 8
13.f odd 12 1 inner 117.2.ba.b ✓ 8
13.f odd 12 1 1521.2.i.c 8
13.f odd 12 1 1521.2.i.f 8
39.h odd 6 1 1521.2.i.c 8
39.i odd 6 1 1521.2.i.f 8
39.k even 12 1 inner 117.2.ba.b ✓ 8
39.k even 12 1 1521.2.i.c 8
39.k even 12 1 1521.2.i.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.ba.b ✓ 8 1.a even 1 1 trivial
117.2.ba.b ✓ 8 3.b odd 2 1 inner
117.2.ba.b ✓ 8 13.f odd 12 1 inner
117.2.ba.b ✓ 8 39.k even 12 1 inner
1521.2.i.c 8 13.e even 6 1
1521.2.i.c 8 13.f odd 12 1
1521.2.i.c 8 39.h odd 6 1
1521.2.i.c 8 39.k even 12 1
1521.2.i.f 8 13.c even 3 1
1521.2.i.f 8 13.f odd 12 1
1521.2.i.f 8 39.i odd 6 1
1521.2.i.f 8 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - T_{2}^{4} + 1 \) acting on \(S_{2}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 194T^{4} + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} - 4 T^{3} + 20 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 16T^{4} + 256 \) Copy content Toggle raw display
$13$ \( (T^{2} + 7 T + 13)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + 12 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( (T^{4} + 8 T^{3} + 20 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 48 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$29$ \( T^{8} - 76 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$31$ \( (T^{4} + 8 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 22 T^{3} + \cdots + 5329)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 120 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$43$ \( (T^{2} - 6 T + 12)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + 17696 T^{4} + 7311616 \) Copy content Toggle raw display
$53$ \( (T^{4} + 76 T^{2} + 121)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 144 T^{6} + \cdots + 116985856 \) Copy content Toggle raw display
$61$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{3} + 20 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 96 T^{6} + \cdots + 3748096 \) Copy content Toggle raw display
$73$ \( (T^{4} - 22 T^{3} + \cdots + 2209)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 104)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 1249198336 \) Copy content Toggle raw display
$89$ \( T^{8} - 384 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$97$ \( (T^{4} + 14 T^{3} + \cdots + 484)^{2} \) Copy content Toggle raw display
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